1) 6x^2 - 12 = 0
6(x^2 - 2) = 0
x^2 = 2
x = ±√2
2) 3a^2 + 5a + 2 = 0
D = b^2 - 4ac = 25 - 4 * 3 * 2 = 25 - 24 = 1, √D = 1.
a1 = (-5 + 1) / 6 = -4/6 = -2/3
a2 = (-5 - 1) / 6 = -1
3) 4x + 4x^2 + 1 = 0
4x^2 + 4x + 1 = 0
D = k^2 - ac (вторая формула для нахождения дискр.) = 2^2 - 4*1 = 0
x1 = -2 + 0 / 4 = -0,5
4) 3x^2 + 7x - 6 = 0
D = b^2 - 4ac = 49 - 4 * 3 * (-6) = 49 +72 = 121, √D = 11
x1 = -7 + 11 / 6 = 4/6 = 2/3
x2 = -7 - 11 / 6 = -3
5) 5x^2 - 22x - 15 = 0
D = k^2 - ac = 11^2 - 5 * (-15) = 121 + 75 = 196, √D = 14
x1 = 11 + 14 / 5 = 25 / 5 = 5
x2 = 11 - 14 / 5 = -3/5 = - 0,6
6) 3x^2 - 10x + 9 = 0
D = k^2 - ac = 25 - 3 * 9 = 25 - 27 = -2, √D < 0, корней нет.
b)
3
x
+3
x+2
<270
3
x
+3
2
∗3
x
<270
3
x
+9∗3
x
<270
10∗3
x
<270 ∣:10
3
x
<27
3
x
<3
3
x<3.
ответ: x∈(-∞;3).
h)
\4*4^x-2\geq 7*2^x\\4*(2^2)^x-7*2^x-2\geq 0\\4*2^{2x}-7*2^x-2\geq 0\\\
4∗4
x
−2≥7∗2
x
4∗(2
2
)
x
−7∗2
x
−2≥0
4∗2
2x
−7∗2
x
−2≥0
Пусть 2ˣ=t ⇒
\4t^2-7t-2\geq 0\\4t^2-8t+t-2\geq 0\\4t*(t-2)+(t-2)\geq 0\\(t-2)*(4t+1)\geq 0\\(2^x-2)*(4*2^x+1)\geq 0\\4*2^x+1 > 0\ \ \ \ \Rightarrow\\2^x-2\geq 0\\2^x\geq 2\\2^x\geq 2^1\\x\geq 1.\
4t
2
−7t−2≥0
4t
2
−8t+t−2≥0
4t∗(t−2)+(t−2)≥0
(t−2)∗(4t+1)≥0
(2
x
−2)∗(4∗2
x
+1)≥0
4∗2
x
+1>0 ⇒
2
x
−2≥0
2
x
≥2
2
x
≥2
1
x≥1.
ответ: x∈[1;+∞).
